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• A B1A003 Total No. of pages:2

Page 1 of 2

Reg. No._______________ Name:__________________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, DEC 2016

Course Code: MA201

Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration:3. Hours

PART A

1.a Show that � = �� − 3���is harmonic and hence find its harmonic conjugate. (8)

b Find the image of �� − �

� � ≤

� under the transformation =

� . Also find the fixed points

of the transformation � = �

� (7)

2.a Define an analytic function and prove that an analytic function of constant modulus is

constant. (8)

b Find the linear fractional transformation that maps �� = 0, �� = 1, �� = ∞onto

�� = −1, �� = −�, �� = 1 respectively. (7)

3.a Show that �(�) = ������� − �������� is differentiable everywhere. Find

its derivative. (8)

b Find the image of the lines � = � and � = �, where �&�are constants, under the

transformation � = ����. (7)

PART B

4.a Evaluate ∫ �� (�) �� �

where � is a straight line from 0 to 1 + 2�. (7)

b Show that ∫ ��

���� =

�√�

� (8)

5.a Integrate ��

���� counterclockwise around the circle |� − 1 − �| =

� by Cauchy’s

Integral Formula. (7)

b Evaluate ∫ ����

������� ��

� where � is |� − 2 − �| = 3.5 by Cauchy’s Residue Theorem

(8)

6.a If �(�) = �

�� find the Taylor series that converges in |� − �| < �and the Laurent’s

series that converges in |� − �| > �. (8)

b Define three types of isolated singularities with an example for each. (7)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

• A B1A003 Total No. of pages:2

Page 2 of 2

PART C

7.a Solve by Gauss Elimination:

�� − �� + �� = 0,

−�� + �� − �� = 0,

10 �� + 25 �� = 90,

20 �� + 10 �� = 80. (5)

b Find the rank. Also find a basis for the row space and column space for

� 0 1 0 −1 0 −4 0 4 0

� (5)

c Find out what type of conic section the quadratic form

� = 17 �� − 30 �� + 17 �� = 128 represents and transform it to the principal

axes. (10)

8.a Find whether the vectors [1 2−1 3], [2 −13 2]��� [−1 8−9 5] are

linearly dependent. (5)

b Show that the matrix � = � 1 2 2 −2

� is symmetric. Find the spectrum. (5)

c Diagonalise � = � 8 −6 2 −6 7 −4 2 −4 3

� (10)

9. a. Determine whether the matrix

⎣ ⎢ ⎢ ⎡ 1 0 0

0 1 √2

� −1 √2

0 1 √2

� 1 √2

� ⎦ ⎥ ⎥ ⎤ is orthogonal? (5)

b. Find the Eigen values and Eigen vectors of � 1 1 2 −1 2 1 0 1 3

� (5)

c. Define a Vector Space with an example. (10)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

• A B3A005 Pages:2

Page 1 of 2

Reg. No._____________ Name:_____________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, MARCH 2017

MA 201: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration: 3 Hours

PART A

1. a. Check whether the following functions are analytic or not. Justify your answer.

i)   zzf z (4)

ii)   2zzf 

(4)

b. Show that   zzf sin is analytic for all z. Find  zf  (7)

2. a. Show that 323 yyxv  is harmonic and find the corresponding analytic function

     yxivyxuzf ,,  (8)

b. Find the image of 10  x , 1 2

1  y under the mapping zew  (7)

3. a. Find the linear fractional transformation that carries �� = −2, �� = 0 and �� = 2

on to the points �� = ∞, �� = 1

4� and �� = 3

8� . Hence find the image of x-axis.(7)

b. Find the image of the rectangular region   x , bya  under the mapping

zw sin (8)

PART B

4. a. Evaluate ∫ |�|�� �

where

i) C is the line segment joining -i and i (3)

ii) C is the unit circle in the left of half plane (4)

b. Verify Cauchy’s integral theorem for �� taken over the boundary of the rectangle

with vertices -1, 1, 1+i, -1+i in the counter clockwise sense. (8)

5. a. Find the Laurent’s series expansion of   21

1

z zf

  which is convergent in

i) |� − 1| < 2 (4)

ii) |� − 1| > 2 (4)

b. Determine the nature and type of singularities of

i) 2

2

z

e z (3)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

• A B3A005 Pages:2

Page 2 of 2

ii) � sin (� � )

(4)

6. a. Use residue theorem to evaluate    

dz zz

zz

C

  

1312

52330 2

2

where C is 1z (7)

b. Evaluate  

dx x

 

0 221

1 using residue theorem. (8)

PART C

7. a. Solve the following by Gauss elimination

y + z – 2w = 0, 2x – 3y – 3z + 6w = 2, 4x + y + z – 2w = 4 (6)

b. Reduce to Echelon form and hence find the rank of the matrix

  

  



1502121

5424426

2203

(6)

c. Find a basis for the null space of

  

  

 

402

840

022

(8)

8. a. i) Are the vectors (3 -1 4), (6 7 5) and (9 6 9) linearly dependent or

ii) Is all vectors  zyx ,, in ℝ� with 04  zxy form a vector space over the field

b. i) Find a matrix C such that xCxTQ  where

2 331

2 221

2 1 5243 xxxxxxxQ  (4)

ii) Obtain the matrix of transformation

y1 = cos θ x1 – sin θ x2, y2 = sin θ x1 + cos θ x2

Prove that it is orthogonal. Obtain the inverse transformation. (6)

9. a. Find the eigenvalues, eigenvectors and bases and dimensions for each Eigen space

of

  

  





021

612

322

A

(10)

b. Find out what type of conic section, the quadratic form 128173017 2221 2 1  xxxx

and transform it to principal axes. (10)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

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